A100452 Triangle read by rows, based on array described below.

1, 3, 4, 7, 8, 9, 13, 14, 15, 16, 19, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 39, 40, 42, 44, 45, 48, 49, 49, 50, 51, 52, 55, 60, 63, 64, 63, 64, 66, 68, 70, 72, 77, 80, 81, 79, 80, 81, 84, 85, 90, 91, 96, 99, 100, 91, 92, 93, 96, 100, 102, 105, 112, 117, 120, 121

Offset

1,2

Comments

The interesting property of this array is that the main diagonal gives A000960.

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

H. Killingbergtro and C. U. Jensen, Problem 116, Nord. Mat. Tidskr. 5 (1957), 160-161.

Formula

Form an array a(m,n) (n >= 1, 1 <= m <= n) by: a(1,n) = n^2 for all n; a(m+1,n) = (n-m)*floor( (a(m,n)-1)/(n-m) ) for 1 <= m <= n-1.

Example

Array begins:
  1 4 9 16 25 36 49 64 81 100 ...
    3 8 15 24 35 48 63 80  99 ...
      7 14 21 32 45 60 77  96 ...
        13 20 30 44 55 72  91 ...
           19 28 42 52 70  90 ...
and triangle begins:
   1
   3  4
   7  8  9
  13 14 15 16
  19 20 21 24 25
  27 28 30 32 35 36
  ...

Mathematica

max=11; a[1, n_]:= n^2;
a[m_, n_]/; 1<m<=n := a[m, n]= (n-m+1)*Floor@((a[m-1, n] -1)/(n-m+1));
a[_, _]=0;
t= Table[a[m, n], {m, max}, {n, m, max}];
Flatten[Table[t[[m-n+1, n]], {m, max}, {n, m}]] (* Jean-François Alcover, Feb 21 2012 *)

Prog

(Magma)
function t(n, k) // t = A100452
  if k eq 1 then return n^2;
  else return (n-k+1)*Floor((t(n, k-1) -1)/(n-k+1));
  end if;
end function;
[t(n, n-k+1): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 07 2023
(SageMath)
def t(n, k): # t = A100452
    if (k==1): return n^2
    else: return (n-k+1)*((t(n, k-1) -1)//(n-k+1))
flatten([[t(n, n-k+1) for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Apr 07 2023

Crossrefs

Cf. A000960, A100453.

Column sums give A100454.

Row 1 = A000290, row 2 = A000290 - 1, row 3 = A100451.

See also A100461.

Sequence in context: A154366 A226227 A319736 * A004201 A109054 A350690

Adjacent sequences: A100449 A100450 A100451 * A100453 A100454 A100455

Keyword

nonn,tabl,nice

Author

N. J. A. Sloane, Nov 22 2004

Status

approved